Estimators and confidence intervals for plant area density at voxel scale with T-LiDAR

Abstract

Terrestrial LiDAR becomes more and more popular to estimate leaf and plant area density.Voxel-based approaches account for this vegetation heterogeneity and significant work has been done in this recent research field, but no general theoretical analysis is available.Although estimators have been proposed and several causes of biases have been identified, their consistency and efficiency have not been evaluated. Also, confidence intervals are almost never provided.In the present paper, we solve the transmittance equation and use the Maximum Likelihood Estimation (MLE), to derive unbiased estimators and confidence intervals for the attenuation coefficient, which is proportional to leaf area density. The new estimators and confidence intervals are defined at voxel scale, and account for the number of beams crossing the voxel, the inequality of path lengths in voxel, the size of vegetation elements, as well as for the variability of element positions between vegetation samples. They are completed by numerous numerical simulations for the evaluation of estimator consistency and efficiency, as well as the assessment of the coverage probabilities of confidence intervals.• Although commonly used when the beam number is low, the usual estimators are strongly biased and the 95% confidence intervals can be ≈ ±100% of the estimate.

Accounting for Wood, Foliage Properties and Laser Effective Footprint in Estimations of Leaf Area Density from multiview-LiDAR Data

“…Estimator for the LAD with multiview-LiDAR data in volumes of interest, which naturally extends the formulation presented in Pimont et al (2018) to actual field data, with the presence of wood volumes, wood hits, correction terms to account for beam divergence and vegetation clumping, as well as to multiview data. The method is applied to an example virtual vegetation scene, and is compared to other common techniques used to combine information from different viewpoints, presented in Appendix C for brevity.…”

“…Here, we briefly summarize the PAD estimation in the mathematical framework proposed by Pimont et al (2018), in which a correction factor was included to account for the effective footprint in clumped vegetation (Soma et al 2018). This factor H varies with the distance of measurement and the voxel size.…”

“…(2011). Such estimator is biased when the beam number is low or when vegetation elements are not infinitely small and can be corrected with a more sophisticated estimator , referred to as the theoretically-bias corrected MLE (TBC-MLE, Pimont et al 2018. In this estimator, each free path is replaced by the effective free path " :…”

“…A; < ranges between 0 and 1). The second term in brackets corresponds to the bias-correction term suggested in Pimont et al (2018), which can be neglected when the beam number is high (i.e. larger than 30).…”

“…other methods applied to voxels or tree crowns (Béland et al 2011;Béland et al 2014a;Hu et al 2018;Yan et al 2019). Also, the theoretical formulation presented in Pimont et al (2018) neglects the presence of woody elements in the estimation of LAD, which should be accounted for separately, either using a separation between leaf and wood returns (Béland et al 2011;Béland et al 2014a) or "leaf-off" scans (Soma et al 2018;Hu et al 2018). To date, a theoretical framework for such inclusion is still missing.…”